shows an interesting asymptotic behavior of the correction which falls off
more slowly than the classical curvature correction \(-2M/x\)  of the lapse
function. Using this term as a correction of Newton's potential in a
weak-field line element shows that non-local effects can imply larger
corrections than effective field theory in a derivative expansion, where the
leading correction would be of the order \(1/x^3\) 
[1]}. However, our simplified solution (REF ), based
on the \(E\) -term in (REF ), does not apply in the completely asymptotic
regime where the \(U\) -term in (REF ) would be dominant. Since this term
implies an asymptotic behavior of \(\delta \phi _2\sim \sqrt{U}x\) , the
corresponding \(\delta N\)  according to (REF ) is \(\delta N\sim \sqrt{U} \log (2M/x)\) . Interpreted as a correction to Newton's potential, this
term suggests a relationship with infrared contributions, consistent with the
interpretation of \(U\) -like terms in cosmological models
[2]}, [3]}.
